Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Real part of $(1+2i)^n$

Is it true that for all $n\in \mathbb{N}$, $n\ge 2$ we have $$|\textrm{Re}((1+2i)^n)|>1?$$ I do know de Moivre's Theorem. I do not know how to show that $|\sqrt{5}^n\cos(n\arccos\left ( \frac{1}{5} \right ))|>1$ because the value…
Tzara_T'hong
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Why imaginary numbers axis is plotted perpendicular to the real numbers axis?

Negative numbers axis is plotted to the opposite side of the positive real number axis that make sense but i do not understand why imaginary numbers are plotted perpendicular to the real numbers axis.
hur chu chu
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For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$

If $|z|=1$, show that: $$\mathrm{Re}\left(\frac{1 - z}{1 + z}\right) = 0$$ I reasoned that for $z = x + iy$, $\sqrt{x^2 + y^2} = 1\implies x^2 + y ^2 = 1$ and figured the real part would be: $$\frac{1 - x}{1 + x}$$ I tried a number of…
stariz77
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When does $(e^a)^b = e^{ab}$ hold?

For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A? Since $e^{2\pi i} = 1$, if this formula holds for all complex numbers $A$ and real numbers $B$, then…
FeelStupid
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Why does the imaginary number $i$ satisfy $i\times 0=0$?

Why does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that.
user161336
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How to show that $\overline{zw}=\overline{z}\,\overline{w}$?

I thought about first multiplying two complex which aren't in the conjugate form: $$zw=a c+i a d+i b c-b d$$ Then multiply two complex conjugates: $$\overline{z}\,\overline{w}=a c\color{red}{-}i a d\color{red}{-}i b c-b d$$ And then note the…
Red Banana
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How to prove $\prod _{a=0}^{9}\prod_{b=0}^{100}\prod_{c=0}^{100}(w^a+z^b+z^c)\equiv?\pmod {101}$

Let $z=e^{\dfrac{2\pi i}{101}},w=e^{\dfrac{2\pi i}{10}}$,show that $$A=\prod _{a=0}^{9}\prod_{b=0}^{100}\prod_{c=0}^{100}(w^a+z^b+z^c)$$ find $A\equiv ?\pmod {101}$ I guess this problem maybe is USA problem,but I can't find it,and I can't solve…
math110
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Law of exponent for complex numbers

Why does the following law $$(z_1z_2)^w=z_1^wz_2^w$$ sometimes fail, such as when $z_1=z_2=-1$ and $w=-i$? And what are the conditions I need for $z_1$ and $z_2$ so that the above identity hold for all complex values of w?
The First StyleBender
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Impossibility of ordering the complex numbers

I have to exposition about the impossibility of ordering the complex numbers: Axioms $6$: Exactly one of the relations $x = y$, $x < y$, $x > y$ holds. Axioms $7$: If $x < y$, then for every z we have x + z < y + z. Axioms $8$: If $x > y$ and $y >…
Yobani Ordoñez
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Real and Imaginary Parts of tan(z)

This is where I'm at: I know $$ \cos(z) = \frac{e^{iz} + e^{-iz}}{2} , \hspace{2mm} \sin(z) = \frac{e^{iz} - e^{-iz}}{2i}, $$ where $$ \tan(z) = \frac{\sin(z)}{\cos(z)}. $$ Applying the above, with a little manipulation, gives me: $$ \tan(z) =…
Garland
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Given complex $|z_{1}| = 2\;\;,|z_{2}| = 3\;\;, |z_{3}| = 4\;\;$ : when and what is $\max$ of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$

If $z_{1},z_{2},z_{3}$ are three complex number Such that $|z_{1}| = 2\;\;,|z_{2}| = 3\;\;, |z_{3}| = 4\;\;$ Then $\max$ of $|z_{1}-z_{2}|^2+|z_{2}-z_{3}|^2+|z_{3}-z_{1}|^2$ $\bf{My\; Try::}$ Let $z_{1}=2\left(\cos \alpha+i\sin \alpha\right)$ and…
juantheron
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Complex Numbers $\stackrel{?}{=} \mathbb{R}^ 2$

Suppose we have a vector field over real numbers $\mathbb R^2$. In additon to vector field proporties define inner product $(x,y) = x_1\cdot y_1 + x_2\cdot y_2$, where $x_1,x_2,y_1,y_2$ are real numbers.This structure is called $2$-dimensional…
user247511
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3 answers

Formula for raising a complex number to a power

Is there an existing formula to raise a complex number to a power? That is, I want to compute $(1 + i)^N$. I basically want to write a function like so: def raiseComplexNumberToPower(float real, float imag, float power): return // (real + imag)…
akaitora
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How to calculate $i^i$

I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python a**b = $a^b$ ) and found out the answer to be a real number with value $0.2079$, How to calculate this value of $i^i$?
Pruthvi Raj
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The $n$ complex $n$th roots of a complex number $z$

Suppose $z$ is a nonzero complex number, so $z=re^{i\theta}$. Show that there are only $n$ distinct complex $n$-th roots, given by $r^{1/n}e^{i(2\pi k+\theta)/n}$ for $0\leq k\leq n-1$. My proof: Let $z=re^{i\theta}\in\mathbb{C}$ and…
Emir
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